Consider now a holomorphic automorphism \(\varphi \)  of \(\Sigma \) . We will be mostly be interested in the case when \(g\ge 2\) , so that \(\varphi \)  automatically has finite order; if the case of a surface of genus 1 (which corresponds to the computations of flat three-manifolds in [1]}) we will take this as an assumption. Then \(\varphi \)  acts on the set of spin structures on \(\Sigma \)  by pull-back with at least one fixed point [2]}. Given such a fixed point \(\mathfrak {s}\) , \(\varphi \)  lifts to an automorphism of the spin structure \(\tilde{\varphi }\) ; if \(\varphi \)  has order \(e\) , then \(\tilde{\varphi }^e\)  is a lift of the identity hence it is either \(\pm \mathrm {id}\) . This leads in general dimensions to a quite rich story, see [3]}, [4]}; in the case of surfaces, the situation is quite simple, and we record some results we will use later.
